| x | P(x) | x * P(x) |
|---|---|---|
| 0 | 0.10 | 0.00 |
| 1 | 0.20 | 0.20 |
| 2 | 0.30 | 0.60 |
| 3 | 0.25 | 0.75 |
| 4 | 0.15 | 0.60 |
Analysis of Variance
Department of Educational Psychology
Agenda
1 Overview and Introduction
2 Probability Distribution Function (PDF) for a Discrete Random Variable
3 Mean or Expected Value and Standard Deviation
4 Binomial Distribution
5 Conclusion
We previously described discrete data, or that which is numeric and “count-able”, that has no intervals between integers.
In this lecture, we will introduce the concept of random variables, or those that can vary in subsequent experiments (used in the sense as how it was introduced during the probability lecture)
The notation of random variables is as follows:
Example from book:
Agenda
1 Overview and Introduction
2 Probability Distribution Function (PDF) for a Discrete Random Variable
3 Mean or Expected Value and Standard Deviation
4 Binomial Distribution
5 Conclusion
| x | P(X = x) |
|---|---|
| 0 | 0.10 |
| 1 | 0.20 |
| 2 | 0.30 |
| 3 | 0.25 |
| 4 | 0.15 |
or…
| x | P(x) |
|---|---|
| 0 | P(x = 0) = 10/100 |
| 1 | P(x = 1) = 20/100 |
| 2 | P(x = 2) = 30/100 |
| 3 | P(x = 3) = 25/100 |
| 4 | P(x = 4) = 15/100 |
Agenda
1 Overview and Introduction
2 Probability Distribution Function (PDF) for a Discrete Random Variable
3 Mean or Expected Value and Standard Deviation
4 Binomial Distribution
5 Conclusion
With PDFs, we sometimes may wish to find the expected value, or the “long-term” average or mean. Thus, doing running this experiment over and over again, we’d expect to converge on this expected mean
This is based in the Law of Large Numbers a topic alluded to several times in the previous modules
For a discrete probability function:
\[ \mu = \sum{(x \cdot P(x))} \]
\[ \sigma = \sqrt{\sum{[(x - \mu)^2 \cdot P(x)]}} \]
Car example from earlier:
| x | P(x) | x * P(x) |
|---|---|---|
| 0 | 0.10 | 0.00 |
| 1 | 0.20 | 0.20 |
| 2 | 0.30 | 0.60 |
| 3 | 0.25 | 0.75 |
| 4 | 0.15 | 0.60 |
\[ \mu = \sum{(x * P(x))} = 2.15 \]
Car example from earlier:
| x | P(x) | x * P(x) | (x - mu)^2 * P(x) |
|---|---|---|---|
| 0 | 0.10 | 0.00 | 0.462250 |
| 1 | 0.20 | 0.20 | 0.264500 |
| 2 | 0.30 | 0.60 | 0.006750 |
| 3 | 0.25 | 0.75 | 0.180625 |
| 4 | 0.15 | 0.60 | 0.513375 |
\[ \sigma = \sqrt{\sum{(x - \mu)^2 * P(x)}} = 1.1948 \]
Probability distribution functions serve a purpose - that is - they can describe a particular pattern in the probability of outcomes in the data.
Once we understand what pattern the variable fits, we can use this information for other analyses, as there are the 4 specialty distributions mentioned earlier: geometric, hypergeometric, poisson, and binomial
Agenda
1 Overview and Introduction
2 Probability Distribution Function (PDF) for a Discrete Random Variable
3 Mean or Expected Value and Standard Deviation
4 Binomial Distribution
5 Conclusion
The binomial distribution has a certain, fixed number of trials represented as \(n\)
There are two possible outcomes in the binomial distribution, “success” and “failure”
The experiment described above fits the binomial probability distribution. Where the random discrete variable \(X\) represents the number of successes obtain in \(n\) trials
For the binomial probability distribution:
\[ \mu = n * p \]
\[ \sigma^2 = n * p * q \]
\[ \sigma = \sqrt{n * p * q} \]
\[ X \sim B(n,p) \]
Agenda
1 Overview and Introduction
2 Probability Distribution Function (PDF) for a Discrete Random Variable
3 Mean or Expected Value and Standard Deviation
4 Binomial Distribution
5 Conclusion
In this shorter lecture, we introduced the concept of discrete random variables, and how they can be represented with probability distribution functions
We played around with calculating expected mean and standard deviations for outcomes from the probability distribution function table
We introduced the binomial distribution as a special, specific pattern for a probability distribution represented via “successes” and “failures”
Module 4 Lecture - One-way ANOVA and Multiple Comparison Procedures || Analysis of Variance